Simple Loading Paths Research Articles

Forming limit of AZ31B magnesium alloy under in-plane complex biaxial loading paths with cruciform specimens

This paper focused on the forming limit curves (FLC) of the AZ31B magnesium alloy sheet metal under different complex loading paths in-plane. Compared with simple loading paths, the study of forming limit of sheet metal under complex loading paths was more practical. In this study, biaxial tensile test of the cruciform specimen was utilized to acquire the FLC after the cruciform specimen was subjected to uniaxial pre-tensile process along rolling direction, which experimentally characterize the forming limits under different pre-strain values of 1.15%, 2.34%, 4.0%, and 5.4%. Subsequently, based on the constitutive model, Hill48 yield criterion and shear failure criterion, the theoretical FLCs of AZ31B magnesium alloy sheet metal was be calculated. Meanwhile, the normal biaxial tensile tests were also simulated by Abaqus to obtain the limit strains by Finite Element Analysis (FEA). Extracting limit strains from the fracture region in FEA model when the first derivative of the strain difference with respect to time reached the maximum value between the first principal strain of necked element and the first principal strain of the element adjacent to the necked zone. As a result, the experimental FLCs was qualitatively consistent with the result of theoretical model and FEA model. In addition, the FLCs of AZ31B magnesium alloy sheet moved to the upper left of forming limit diagram with the increase of the pre-strain, which indicated that the pre-strain process along the rolling direction could significantly improve the formability of the sheet metal.

Investigation on a full coupling between damage and other thermomechanical behaviours in the standard thermodynamic framework including environmental effects

In the present paper, constitutive equations accounting for coupled damage-thermo-elasto-(visco)plastic and diffusion at small deformation are proposed in the standard thermodynamics of irreversible processes framework. One main objective is to include the diffusion phenomenon in the models with ductile damage. The model is developed in the framework of thermodynamics of irreversible processes with a set of internal state variables. For illustration, we consider hydrogen diffusion in both normal interstitial lattice sites and trapping sites. The damage modelling of microvoids or microcracks is introduced by the use of Continuum Damage Mechanics framework leading to the definition of effective state variables on fictive undamaged configuration based on the total energy equivalence assumption. Consequently, the full coupling concerns not only the elastic and inelastic behaviour with hardening (isotopic and kinematic), but also the thermal and diffusion phenomena. The concept of the total energy equivalence is thus extended to define effective temperature, entropy, and effective state variables associated with diffusion. It enables to obtain different couplings between thermal phenomena, diffusion phenomena, and the mechanical behaviour, especially isotropic ductile damage. Such a full coupled model is then applied to a representative volume element subject to some typical simple loading paths for illustration and test of such couplings.

Undrained response of Sydney sand under non-reversal cyclic loading

Stress history and loading path can significantly influence the cyclic response of sands. It is well known that cyclic behaviour under non-symmetrical cycling is generally different from that under symmetrical loading around zero deviator stress. Anisotropic consolidation prior to cycling in triaxial apparatus changes the behaviour by exerting an initial static shear stress on critical planes. The present paper reports the results from a number of cyclic triaxial tests on anisotropically consolidated samples of Sydney beach sand. The samples are reconstituted in the laboratory and subjected to non-reversal cycling in compression only loading under undrained conditions. Typical test behaviours are described and the effects of cyclic stress ratio and initial state on the response are discussed. It is shown that the state parameter is capable of predicting the cyclic resistance and the trend of excess pore water pressure generation during one-way compressional cycling and, as has been well established for simple loading paths, critical state soil mechanics is able to provide a reliable framework to characterize the behaviour under different cyclic loading conditions.

Modeling of the induced plastic anisotropy fully coupled with ductile damage under finite strains

The introduction of the kinematic hardening induced distortion of the yield surface is expected to enhance the capability of the model in simulating the material behavior under complex loading paths occurring during sheet metal forming processes.In this work non-associative elastoplastic anisotropic constitutive equations accounting for mixed nonlinear hardening and fully coupled with isotropic ductile damage, is improved by considering the distortion of the yield subsequent surfaces. A convenient “distortion” deviatoric stress tensor S̲d is used in both yield criterion and plastic potential instead of the usual classical deviatoric stress tensor S̲. Accordingly, the yield surface distortion is controlled by three adequate material parameters named Xl1c, Xl1p and Xl2c. A parametric study is conducted under simple loading paths (tension and torsion) in order to explore the capabilities of the model in describing the yield surface distorsion. Comparisons with experimental results on aluminum alloy, taken from the literature, are also given. Moreover, the influence of the subsequent yield surfaces distortion under complex non-proportional loading paths on the evolution of the plastic flow with hardening and the ductile damage are also discussed and highlighted.

Analytical and numerical analysis of frictional damage in quasi brittle materials

Frictional sliding and crack growth are two main dissipation processes in quasi brittle materials. The frictional sliding along closed cracks is the origin of macroscopic plastic deformation while the crack growth induces a material damage. The main difficulty of modeling is to consider the inherent coupling between these two processes. Various models and associated numerical algorithms have been proposed. But there are so far no analytical solutions even for simple loading paths for the validation of such algorithms. In this paper, we first present a micro-mechanical model taking into account the damage-friction coupling for a large class of quasi brittle materials. The model is formulated by combining a linear homogenization procedure with the Mori–Tanaka scheme and the irreversible thermodynamics framework. As an original contribution, a series of analytical solutions of stress–strain relations are developed for various loading paths. Based on the micro-mechanical model, two numerical integration algorithms are exploited. The first one involves a coupled friction/damage correction scheme, which is consistent with the coupling nature of the constitutive model. The second one contains a friction/damage decoupling scheme with two consecutive steps: the friction correction followed by the damage correction. With the analytical solutions as reference results, the two algorithms are assessed through a series of numerical tests. It is found that the decoupling correction scheme is efficient to guarantee a systematic numerical convergence.

Evaluation on strength and deformation behavior of red sandstone under simple and complex loading paths

Triaxial compression experiments were carried out for red sandstone to investigate its strength and deformation failure behavior under simple and complex loading paths. In this research, tested confining pressure is in the range from 5 to 35MPa. Under simple loading path, the peak strength, residual strength and critical damage threshold of red sandstone all increase with the confining pressure, which are in good agreement with the linear Mohr–Coulomb criterion. To investigate the re-fractured mechanical behavior of pre-cracked red sandstone with different post-peak stress drops, two complex loading paths with only one specimen are put forward to obtain the strength and deformation parameters of red sandstone. The differences of strength and deformation parameters of red sandstone between simple and complex loading paths are evaluated detailed. The elastic moduli of red sandstone under simple and complex loading paths all increase nonlinearly with the confining pressure. It is suggested and recommended to predict the peak strength of rock under simple loading path by adopting complex loading path A (increasing gradually the confining pressure) not complex loading path B (reducing gradually the confining pressure). On the basis of the linear Mohr–Coulomb criterion, a kind of new method is put forward to revise the peak strength of red sandstone under complex loading path A, which is testified to be right and reasonable. The investigated conclusions are very significant for ensuring the stability and safety of deep underground rock engineering.

Deformations and liquefaction of granular soils in plane strain conditions

Abstract An incremental model is proposed to describe pre-failure behaviour of granular soils in plane strain conditions. The model has been derived from an approach applied in the analysis of triaxial tests. We have applied the methods of tensor algebra to transfer our triaxial results to a 2D case. The model describes the pre-failure deformations of fully drained sands and phenomena associated with undrained response such as static liquefaction. This model has a simple structure in the form of incremental equations. For some simple loading paths, these equations can be integrated analytically. For more complex loading paths a simple numerical algorithms will be sufficient. The model takes into account an initial anisotropy of soil which is ignored in most models. It also takes into account the initial state of soil, defined as rather contractive or dilative, in addition to classical geotechnical division into loose and dense sands. The proposed constitutive equations allow the study of pre-failure instabilities of sand, and some examples are presented.We have also derived analytically the equation for an instability line.

Discussion: “Yield Function for Solder Elastoviscoplastic Modeling” (Dube, M., and Kundu, T., 2005, ASME J. Electron. Packag., 127, pp. 147–156)

Discussion: “Yield Function for Solder Elastoviscoplastic Modeling” (Dube, M., and Kundu, T., 2005, ASME J. Electron. Packag., 127, pp. 147–156)

Investigation on the Formability of Tube in Hydroforming with Radical Crushing under Simple Loading Paths

Tube hydroforming with radial crushing (THFRC) process is particularly applicable to the tube which is difficult to shape due to lack of axial feeding. In this paper, the formability of the circular tube expanded into a triangle cross-section under the simple loading paths is explored by using the numerical simulation. The effect of the forming mode and the loading paths on bulged profile, wall-thickness distribution and the potential fracture location of the bulged tube are analyzed. The results showed that constraint conditions at tube ends have small influence on the bulged profile and wall-thickness distribution. Moreover, the larger the peak value of the internal pressure is, the better material filling ability and higher accurate the tube bulged profile are. Furthermore, the higher forming accuracy, the better material filling ability and wall-thickness distribution are gained in THFRC than those in free hydro-bulging (FHB) process, and they can be obtained under constant than linear pressure loading paths. Finally, the potential fracture location of the bulged tube in THFRC process is quite different from that in FHB process.

Hardening description for FCC materials under complex loading paths

The present work aims at exploring self and latent hardening for FCC polycrystals under complex loading paths at room temperature. Combinations of simple loading paths sequences, such as tension and simple shear, with different orientations with regard to rolling direction, are considered. Experimental results are compared to finite element computations of polycrystalline aggregates taking into account the material microstructure, and to simulations based on mean field models.

Homogenization for Granular Materials: non reversible behaviour

Using a homogenization framework, this paper proposes a description of the non-reversible behaviour of granular materials. First the basic equations are reviewed and some general elasticity results are presented with a comparison to DEM simulations. The global non-reversible behaviour is introduced by the definition of local conditions of contact stability: loss of contact and sliding. The first consequence is the possible definition of global yield surfaces and thus failure surfaces. These surfaces are compared to the usual plastic criteria. The influence of the different homogenization parameters introduced by the definition of localization and averaging operators is also studied. In the last section, a simulation of non-reversible behaviour is presented. Comparisons to experimental data are made in the case of simple loading paths (isotropic compression, triaxial test) as well as in complex loading histories.